Virginia Water Resources Research CenterVirginia Polytechnic Institute and State University

Bulletin 163

A Study of Infiltration Trenches

C.Y. Kuo. J.L.Zhu, and LA. Dotterd

3 3 3

8 9 S T

Bulletin 163April 1989

A Study of Infiltration Trenches

C.Y. KuoJ.L Zhu

L.A. DollardDepartment of Civil Engineering

Virginia Polytechnic Institute and State University

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VPI-VWRRC-BULL1633.5C

Virginia Water Resources Research CenterVirginia Polytechnic Institute and State University

Blacksburg • 1989

This Bulletin is published with funds provided in part by theU.S. Geological Survey, Department of the Interior,

as authorized by the Water Resources Research Act of 1984.

Contents of this publication do not necessarily reflectthe views and policies of the

United States Department of the Interior, no does mentionof trade names or commercial products constitute their

endorsement or recommendation for useby the United States Government.

Additional copies of this publication, while the supply lastsnay be obtained from the Virginia Water Resources Research Center.

Single copies are provided free to persons and organizationswithin Virginia. For those out-of-state, the charge is $8 a copy.

Payment or purchase order must accompany all orders.

CONTENTS

Ust of Figures v

List of Tables ix

Acknowledgments xi

Abstract xiii

List of Symbols xv

Introduction 1

Governing Equations 5I. Derivation of Governing Equations 5

II. Initial and Boundary Conditions 7

Finite Element Analysis 9I. Formulation 9

II. Isoparametric Elements and Numerical Integration 10III. Finite Difference Approximation in Time 12IV. Solution Procedure 12V. Nodal Flux Computation 13

Finite Element Model Verification 15I. 1 -D Transient Vertical Flow 15

II. 2-D Transient Flow 15III. Steady-state Flow through a Square Embankment 16

Fok's Two-Dimensional Infiltration Model 17

Laboratory Model Design and Construction 21

Experimental Results and Comparison with Numerical Models . . . 25I. Sand Experiments 25

II. Clay Experiments 26III. Loam Experiments 27

Parametric Studies 29I. Soil Properties 20

II. Geometries 30III. Water Tables 31IV. Effects of Sediment Deposition in Trench 31V. Distribution of Water Content 31

in

Conclusions 33

Figures 35

Tables 81

References 85

IV

LIST OF FIGURES

Figure 1Comparison of Finite Element Model withvan Genuchten's Model 37

Figure 2Comparison of Finite Element Model with Selim andKirkham's Model 38

Figure 3Comparison of Finite Element Model withHuyakorn's Model 39

Figure 4Relationship between y/h and Kt/nsh 40

Figure 5Top View of Laboratory Model 41

Figure 6Three-dimensional View of Laboratory Model 42

Figure 7Panels of Tailwater Compartment with Outlets 43

Figure 8Relationship of Meter Readings to Soil Suction(SoilMoisture Equipment Corporation 1985) 44

Figure 9Relationship of Meter Readings to Water Content(SoilMoisture Equipment Corporation 1985) 45

Figure 10Guelph Permeameter (SoilMoisture EquipmentCorporation 1986) 46

Figure 11C Factor for Guelph Permeameter (SoilMoistureEquipment Corporation 1986) 47

Figure 12Grain Size Distribution Curve of Sand 48

Figure 13Placement of Soil Moisture Blocks in Sand 49

Figure 14Water Content versus Meter Readings for Sand 50

Figure 15Initial Moisture Condition for First SandExperiment 51

Figure 16Results of First Sand Experiment (10 and 30 Minutes) 52

Figure 17Results of First Sand Experiment (50 and 100 Minutes) 53

Figure 18Results of Second Sand Experiment (Initial Conditionand 20 Minutes . 54

Figure 19Results of Second Sand Experiment (50 and 100 Minutes) 55

Figure 20Water Content versus Meter Readings for Clay 56

Figure 21Grain Size Distribution Curve of Clay 57

Figure 22Placement of Soil Moisture Blocks in Clay 58

Figure 23Results of First Clay Experiment (37 and 57 Minutes) 59

Figure 24Results of First Clay Experiment (147 and 237 Minutes) 60

Figure 25Results of Second Clay Experiment (39 and 84 Minutes) 61

Figure 26R e s u l t s o f S e c o n d C l a y E x p e r i m e n t ( 1 6 4 a n d 2 2 4 M i n u t e s ) . . . . 6 2

Figure 27Water Content versus Meter Readings for Loam 63

Figure 28Grain Size Distribution Curve of Loam 64

Figure 29Placement of Soil Moisture Blocks in Loam . 6 5

Figure 30Results of Loam Experiment (2 and 10 Minutes) . . . 66

Figure 31Results of Loam Experiment (25 and 60 Minutes) 67

Figure 32G e o m e t r y o f t h e T r e n c h f o r P a r a m e t r i c S t u d i e s . . . . . . . . . . 6 8

Figure 33Inflow Hydrograph to the Trench Prescribed forParametric Studies , 6 9

Figure 34Infiltration Rates for Different Soils 70

VI

Figure 35Water Levels in the Trench for Different Soils 71

Figure 36Effects of Deep/Narrow versus Shallow/Wide Trench onInfiltration Rates and Water Levels in Trenches withConstant Volume (Sandy Loam Soil) 72

Figure 37Effects of Deep/Narrow versus Shallow/Wide Trench onInfiltration Rates and Water Levels in Trenches withConstant Volume (Loam Soil) 73

Figure 38Effects of Long/Narrow versus Short/Wide Trench onInfiltration Rates and Water Levels in Trenches withConstant Volume (Sandy Loam Soil) 74

Figure 39Effects of Long/Narrow versus Short/Wide Trench onInfiltration Rates and Water Levels in Trenches withConstant Volume (Loam Soil) 75

Figure 40Effects of Water Table on Infiltration Rate 76

Figure 41E f f e c t s o f S e d i m e n t D e p o s i t i o n i n T r e n c h o n I n f i l t r a t i o n R a t e . . . . 7 7

Figure 42Long-term Simulation of Infiltration in a Trench 78

Figure 43Movement of Wetting Fronts in Soil Medium as aFunction of Time 79

VII

LIST OF TABLES

Table 1Typical Values of van Genuchten's Model 81

IX

ACKNOWLEDGMENTS

The authors wish to express their appreciation to William Walker, MargaretHrezo, and Diana Weigmann of the Water Center for administration of theproject. Our thanks go, also, to the advisory committee and the technicalreviewers of the manuscript. The work upon which this report is based wassupported by funds provided by the Virginia Water Resources Research Center.

XI

ABSTRACT

Urbanization increases peak flow and total volume of surface runoff as comparedto predeveloped conditions. Infiltration trenches in unsaturated soil, one of thebest management practices employed to control excessive runoff in urbanareas, are examined in this report. A two-dimensional finite element model hasbeen developed to simulate the transient flow in a variably saturated porousmedium. Parameters such as soil properties, water table location, initial soilmoisture condition, trench geometry, and surface runoff hydrograph at thefacility site are specified, and routing is performed to find infiltration rate, waterdepth and storage in the trench, and overflow, if any. A laboratory model has alsobeen used to test the validity of the finite element model and an infiltration modeldeveloped by Fok and his colleagues. Parametric studies have been performedwith the finite element model for loam soil and sandy loam to determine thevarious effects of such parameters. Sands are recommended for use in aninfiltration facility while clays are not. Saturated soil conductivity and the Áparameter from van Genuchten's model are found to have the greatest effect oninfiltration rate. Where geometry is concerned, a wide, shallow trench isrecommended for a given trench length and trench volume. A short, wide trenchis recommended for a given depth and trench volume. The depth of the watertable has been shown to have a greater effect in silt than in sand. There exists alimit for each soil beyond which the water table has no effect on infiltration. Thelimit is about 10feet for sand, 30 feet for loamy sand, and 60feet for sandy loam.

In the design of an infiltration trench, geometries vary considerably and waterdepth in the trench varies with time. The water flux across the bottom and sidesof the trench is an important factor in sizing the trench. The finite element modelhas the capability to change geometries and to calculate the flux and water levelin the trench. It is a useful tool in the design of infiltration facilities.

Key Words: Peak Flow, Runoff, Infiltration, Water Table, Trench Geometry

XIII

LIST OF SYMBOLS

a = well radius for Guelph Permeameter, L

A = gross cross-sectional area through which flow occurs, L2

Ci,C2-C factors for Guelph Permeameter

Cu = coefficient of uniformity

D = soil/water diffusivity, LVT

Dso = median soil particle size, L

E - generalized specific water capacity, L"1

f = volumetric flow rate via sources (or sinks) per unit volume ofporous medium, L/T

h = pressure head, L

he-constant capillary potential head at wetting front, L

ho = constant depth of water on soil surface, L

hT-constant head loss in the transmission zone extrapolatedto the wetting front, L

hw = constant head loss in wetting front, L

hx = horizontal pressure head loss, L

H = piezometric head, L

Hi,H2 = well height for first and second measurements ofGuelph Permeameter, L

K-hydraulic conductivity, L/T

Kij - soil hydraulic conductivity tensor, L/T

Ks = saturated conductivity, L/T

n = porosity

n¡ - unit vector normal to the surface of the control volume

q¡ = vector of fluid velocity relative to the grains, L/T

Q = flow rate, LVT

Qx = horizontal flow rate, LVT

r = number of nodal points in an element

R1R2- steady state rate of fall for Guelph Permeameter, L/T

s = degree of saturation

xv

Se-effective saturation

Ss-specific storage coefficient

t - t ime of infiltration, T

V-gross volume of the element, L3

Vv = volume of voids, L3

Vw = volume of water, L3

w = test function

x = lateral advance of wetting front, L

X - reservoir constant for combined reservoirs in Guelph Permeameter, L2

y = length of wetting from soil surface to wetting front during infiltration, L

Y-reservoir constant for inner reservoir only for Guelph Permeameter, l_2

a - compressibility of the porous medium

yS - compressibility of the fluid

f - to ta l boundary of the element domain, L

À - parameter reflecting various types of soils, L"1

H - parameter reflecting various types of soils

v = parameter reflecting various types of soils

Oe = element domain

0j = linear interpolation functions

p = fluid density, M/L3

az - normal intergranular stress on a horizontal plane, M/LP

6- volumetric water content

d, ~ residual water content

6S ~ saturated water content

XVI

INTRODUCTION

It has been recognized that urbanization is responsible for the increase in bothpeak flow and total volume of surface runoff because of the decrease ininfiltration as compared to predeveloped conditions. In many localities, theprimary goal of stormwater management plans is to maintain, as nearly aspossible, the predevelopment runoff characteristics. To control excessive runoffdue to the increases in the impervious area of a watershed, detention basins,infiltration facilities, porous pavements, swales, marshes, and open spaces arecommonly employed. These control structures, known as urban best manage-ment practices, have been proven to be effective in runoff and nonpoint sourcepollution control.

Infiltration facilities in unsaturated soil are examined in this report. An infiltrationstructure is defined as a dry well, pond, or subsurface trench that is used totemporarily store runoff in a stone-filled reservoir. It is generally used onrelatively small drainage areas such as residential lots, commercial areas,parking lots, and open space areas. Surface runoff flows into the trench andinfiltrates the surrounding soil media slowly. Infiltration facilities offer two majoradvantages in urban stormwater management. In terms of surface runoffcontrol, they attenuate and reduce the peak runoff, reduce or eliminate therunoff volume. In terms of subsurface water management, they increase thelocal soil moisture and water table due to infiltration, and feed water into naturalstreams after the storm to augment the low streamf low. The facilities have greatcapability to trap nonpoint source pollutants associated with the runoff. However,pollutants collected in the facility itself have the potential to contaminate theunsaturated and saturated subsurface zones due to long term percolation.

Infiltration structures have been employed in many parts of the country. Higgs(1978) has designed percolation trenches to handle surface runoff from parkinglots in Northern Virginia. A dry well has been designed and constructed to serveas a rain and snowmelt water catchbasin for a flat building roof and parking areain Canada (Beukeboom 1982). Infiltration basins are used to hold urban stormrunoff in Florida and California (Hantzsche and Franini 1980). McBride andSternberg (1983) at the University of Maryland developed a standardizedmethod for selecting the optimum site for an infiltration facility and a designprocedure for determining the storage volume required to control a knownvolume of runoff. Bouwer's steady state equation (1978) was used in theirformulation. Design specifications and procedures are available (Ericsson andGustafson 1982; South Florida Water Management District 1983; MarylandDepartment of Natural Resources 1984; Virginia Water Control Board 1978;Northern Virginia Planning District Commission 1978, 1987; Schueler 1987).Most of the design manuals deal with rational approaches and empirical

formulas. The principle of the conservation of mass is used in the calculation ofthe volume required for the infiltration structure. The infiltration rate is assumedto be constant for a given type of soil. In fact, the infiltration rate is a function ofthe location of the water table, the initial moisture condition of the soil, the waterdepth in the facility, etc. There is a need to rigorously analyze the infiltrationsfacilities by developing a computer model so that design variables such asdimensions and shape, properties of the aggregates in the facility and thesurrounding soil, water table, surface runoff, infiltration rate, storage volume,detention time, etc., can be studied in detail.

Research on infiltration facilities is limited. Most of the research on infiltrationitself has been conducted in connection with irrigation. The American Society ofAgricultural Engineers (1983) sponsored a conference on advances in infiltrationin which many different methods for estimating infiltration were discussed.Aron (in Poertner 1974) conducted a field experiment on infiltration trenches.This experiment used a sprinkler system to simulate rainfall and the infiltrationtrenches were equipped with observation wells. This experiment was limited inthat only one site could be studied. Laboratory models of infiltration trenches arerare. Fok, Chung, and Clark (1982) used a laboratory model to verify theirinfiltration equations. However, this was not a model specifically for infiltrationtrenches. A study of infiltration trenches conducted by Yim and Sternberg (1984)at the University of Maryland used a laboratory model. The model used concretesand as the porous medium with 3/8-inch gravel as aggregates in the infiltrationtrench. The model was used to test the validity of a theoretical infiltrationequation proposed by Bouwer (1978). The model was also used to evaluate theeffects of sediment accumulation on infiltration through the trench and toestablish a practical method of controlling sediment load. The experimentvalidated Bouwer's equation and recommended a natural granular filter to trapsediments instead of a plastic fabric filter commonly used.

A simple analytical model for infiltration basins was given by Li (1983). Thismodel considers the vertical infiltration only, neglecting the horizontal advanceof the wetting front. Two-dimensional infiltration equations for furrow irrigationexpressed in explicit forms have been developed by Fok and his colleagues (Fokand Hansen 1966; Fok 1967; Fok, Chung, and Clark 1982). Vertical andhorizontal advances of the wetting front are calculated based upon Darcy's lawand the principle of continuity. The cumulative infiltration volume in the model iscalculated by multiplying the porosity of soil, the net increment in the degree ofsaturation, and the wetted volume. The unsteady two-dimensional water flowequation for unsaturated soils was solved numerically with a finite differencemethod using an alternating-direction implicit method by Selim and Kirkham(1974). The water table was considered as a water source in their model.However, a capillary fringe zone above the water table was not considered in

either Fok's model or the finite difference model. There are other computermodels available in dealing with flow in unsaturated zones (Yeh 1987; Kalua-rachchi and Parker 1987; and Huyakorn et al. 1986). They are not specificallyformulated for the study of infiltration facilities. In the design of an infiltrationtrench, fo example, trench geometries vary considerably and water depth in thetrench varies with time. The water flux across the bottom and the sides of thetrench is an important factor in connection with sizing the trench. The computermodel must have the capability to solve this particular problem.

The primary objective of this study is to develop a two-dimensional finite elementmodel to simulate the transient flow of water in a variably saturated porousmedium in order to calculate pressure heads and moisture distributions in thesoil medium surrounding the infiltration facility and fluxes across the boundariesof the facility and the water table. The Galerkin finite element method is used.The advantages of using the finite element method include better computationalefficiency, the ability to accommodate irregular geometries of the facilities andsoil domains, and to yield the derivative type by-products such as flux atboundary points of the facility, which is the infiltration rate into the soil medium.Based on the soil properties, initial soil moisture condition, facility geometry, andsurface runoff hydrograph at the facility site, routing is performed to findinfiltration rate, water depth and storage in the facility, and overflow if any. Fromthis information, the size of the infiltration facility is chosen such that thedesirable runoff detention and retention is attained. A laboratory model has alsobeen designed and constructed to test the validity of the finite element formu-lation and the infiltration model developed by Fok and his colleagues. Theverified computer model has been used to aid the design of infiltration trenches.

GOVERNING EQUATIONS

I. Derivation of Governing Equations

Consider a control volume V in an elastic porous medium having a uniformmoisture content. The control volume will be allowed to deform, but the solid isconsidered incompressible. The following definitions are used:

V = gross volume of element

Vu = volume of voids

n - porosity = Vv/V

Vw = volume of water

s - degree of saturation = Vw/Vv

8 = volumetric water content - Vw/V = ns

Applying the law of mass conservation, the flux of f luid mass across the surfaceof the control volume plus the fluid mass increase or decrease due to sources orsinks within the control volume must be equal to the rate of fluid mass storagewithin the control volume. This can be written in the indiciai form:

¡¡i pfdV-¡¡ Pn¡qidA=-^\\\ nspdV (2.1)

where p is the fluid density, f is the volumetric flow rate via sources (or sinks) perunit volume of porous medium, A is the surface of the control volume, n, is a unitvector normal to the surface of the control volume, q¡ is the vector of fluid velocityrelative to the grains, and i = 1,2,3 is the implied summation convention. Byusing the Green's theorem to integrate the second term of the left-hand side andthe Leibnitz rule to integrate the right-hand side, Equation (2.1 ) becomes

Since lateral constraints on an aquifer prohibit significant horizontal defor-mation, it is assumed the volume element will deform only vertically. Thus, thecompressibility of the granular skeleton is defined as

-AV /VK—^r (2.3)

where (7 z is the normal intergranular stress on a horizontal plane. The use of thedefinition of n and Equation (2.3) yields

- | ^ = -aV-~- (2.4)

assuming the solid volume remains constant, i.e., AV = AVV, we can write

-¿t'T ~ôt"~fi "¥"s=(1~~vL)V~dr = - ( 1~n )a~ã/" (25)

The compressibility of the fluid is defined as the change of fluid density Ap withrespect to pressure change AP and is expressed as

Ap/pAp

Thus, it follows

~~df = Pp~df (2,7)

Substituting Equations (2.4), (2.5), and (2.7) into Equation (2.2), one arrives at

ds , „dp daz ôjpqò f 7 f i .

which is a general equation applicable to both steady and unsteady flow in anelastic medium with a degree of saturation s. When the media is saturated (s = 1and 9s/3t - 0), any change in static pore pressure head must immediatelyproduce a change in the intergranular stress throughout the medium (Eagleson1970). That is A<TZ - - AP=pg Ah, where h is pressure head. Then, the governingequation for saturated flow is

oh d(pq¡)

where Ss = apg ô/n + /îpgô is the specific storage coefficient. When media isunsaturated, compressibility terms of Equation (2.8) are unimportant. Theequation governing unsaturated flow is obtained by dropping the last

two terms on the left-hand side of Equation (2.8) as

dö dh d(pq,)p =——— Y pf \i.\\j)

where dô/dh is the water capacity.

The Darcy's law in the vector form is

q¡ = K¡¡ •""""'"" to 11 \

where Kii is the soil hydraulic conductivity tensor, H - h + X2 is the piezometrichead, and x2 is the vertical coordinate representing elevation. SubstitutingEquation (2.11) into the right-hand side of Equation (2.8) and combiningEquation (2.9) and Equation (2.10), the following expression is derived forisothermal, compressible, laminar flow in an elastic porous medium.

where E is the generalized specific water capacity, which is usually written as E -wSs + d0/dh, with to - 1 for h > 0 (saturated) and u> - 0 for h < 0 (unsaturated).

If the principal directions of the hydraulic conductivity tensor coincide with thoseof coordinates (i.e. K¡¡ - 0 when j # j) and assuming incompressible fluid Ifi =0),the two dimensional governing equation in x - y coordinate system can bederived from Equation (2.12) as

where x and y indicate horizontal and vertical directions. In general, E = E(s) andK •= K(s) are functions of the saturation and the saturation is a function of thepressure head s = s(h). There are a number of models for the soil properties. Forexample, the nonlinear functions to describe water capacity and hydraulicconductivity can be taken to be nonhysteretic and defined by van Genuchten(1980 a) as

^±^ls¡,r (2,4,

-(1-SW (2.15)

with

S.-C1 +(¿I/JUT' (2.16)

where Se is the effective saturation, 8S is the saturated water content, 8, is theresidual water content, Ks is the saturated conductivity, and A, /y and v = 1 /(1 x/y)are parameters reflecting various types of soils.

II. Initial and Boundary Conditions

Initial conditions of pressure head must be specified at every nodal point in thedomain

h(x,y,t) = h{x.y,O) (2.17)

Either head or flux-type boundary conditions must be prescribed at theboundaries of the domain

h(x,y,t) = h(xByBt) {2.18)

or

-nxKx • § • -ny(Ky - g - + Ky) - q(xB yB ,0 (2., 9)

where h(xB,yB,t) and q(xB, yB,t) are the prescribed pressure head and net fluid fluxat the boundary and nx and ny are the components of unit normal vector on theboundary. Notice that h(xB,yB,t) and q(xB,yB,t) could be the known functions oftime or could be determined from the model with which they are coupled. Forexample, given the initial water level in the trench and an input surface runoffhydrograph at the site, the water levels in the trench are always updatedaccording to the changing of storage in the trench with time. In other words, theinfiltration rate (i.e. the flux across the boundary of the trench) is a function ofwater depth in the trench. The water depth in the trench, which is related to thestorage in the trench, influences the infiltration rate. All of these variableschange with time. The dependency of storage on boundary condition is solved bya trial and error procedure. When the total flux is specified, the outflow flux willbe computed based on a trial h(xB,VB,t), then the computed flux is checked againstthe specified flux. Then h(sB,yB,t) is adjusted until the difference between thespecified flux and the computed flux is negligible.

8

FINITE ELEMENT ANALYSIS

I. Formulation

Multiplying Equation (2.13) with a test function w and integrating over anelement domain Qe, one has

which can be further written in the variational form as

where f is the total boundary of the element domain Cfh can be approximated by the following expression

h(x,y,t) = ^Jt/fitficy) (3.3)

where h¡ are values of h at time t and at point (xj,y¡), (p\ are linear interpolationfunctions, and r is the number of the nodal points in an element.

Substituting Equation (3.3) and w = 0¡ into Equation (3.1), one obtains

0: • §[{;(/•<•/«• * } • £

^dx dy - \¿*> ""dy

where

L f / l — ^ ' X X "̂

which is defined as flux across boundary l~e. The finite element formulation ofEquation (3.4) can also be written in the matrix form:

{Fe} (3.6)

where

MJJ= f Etrfjdxdy (3.7)

Fî " J <7„0(c/s - J Ky — dxdy + j f <¿>, dx dy (3.9)

If quadrilateral elements are used, i,j = 1,2,3,4, [M6] and [Ke] are 4 x 4 matricesand {Fe} is a vector of 4 components.

II. Isoparametric Elements and Numerical Integration

To facilitate an accurate representation of irregular domains, the isoparametricelements are used. However, it is difficult to compute the element coefficientmatrices and column vectors directly in terms of the global coordinates x and y,which are used to describe the governing equation. This difficulty can beovercome by introducing an invertibte transformation between a curvilinearelement fie and a master element ñ of simple shape that facilitates numericalintegration of the element equations. The coordinates in the master elementsare chosen to be the natural coordinates (£/;) such that -1 <(£»?) < 1. Considerthe coordinate transformation:

(3.10)

where the element interpolation functions are in natural coordinates, h¡ is thesolution at i th node of an element, and (xi,yi) are the global coordinates of the i thnode of element Qe

7(1 +«O-i)

7(1-«(1+1)

(3.11)

10

In order to perform element calculations, one must transform functions of x andy to functions of f and 77. The coordinate transformation of the integrals shouldbe employed.

Let [J] be the Jacobian matrix

õx ày

dx ¿y

and [J]"1 be the inverse of the Jacobian matrix

J21

(3.12)

(3.13)

In Equation (3.8) the integral is a function of x and y. Suppose that the mesh offinite elements is generated by a master element ft. Using the transformation,the following equation is derived.

A

0*1

(3.14)

Now the integral is defined over a rectangular master element and thequadrature can be presented in this form:

L = f' (fJ-iJ-i

ff (3.15)

where M and N denote the number of the quadrature points in the f and t)directions, (£\,r)j) denote the Gauss points, and Wi and Wj denote the corre-sponding Gauss weights. Mass matrix M¡¡ and force vector F| are evaluated in thesame way. Details involved in the numerical evaluation of element matrices hasbeen shown by Reddy (1984).

11

III. Finite Difference Approximation in Time

Finite difference methods are typically used in the approximation of the timederivative. Equation (3.6) is approximated by a forward difference scheme as:

[Ke]{rty+1 = (3.16)

where

IV. Solution Procedure

Based on the inter-element continuity conditions, all the element matrices areassembled into a global finite model, which is in general a system of equations inthe form:

A A

M2 A13

K2 1 K;A A

Kol K

A

Kn

23A'

23 K33

A

A

A

Kn

X

^3

•

^3

• (3.18)

This system of equations is solved by Gaussian elimination for values of h¡k+1 attime level k + 1. The nonlinearity (dependency of coefficient matrices on thesolution) can be solved by an iterative method. The solution is attempted by aprocess of successive approximations. This involves making an initial guess andthen improving the guess by some iterative process until an error criterion issatisfied.

An iterative scheme based on Newton-Raphson's method has been developed.Using indiciai notation with the summation convention implied, Equation (3.18)can be written as

- F¡ = R, ij = 1,2,3, ....n. (3.19)

12

where h¡ is the approximation in current iteration and R¡ is the residual. Tominimize the residual, the improvement AH¡ is found through

K¡ Ahj = -R¡ (3.20)

where KT¡¡ is the tangent or Jacobian matrix which is derived from Equation(3.19) as

ur_ àRt _u õhk ÕKlk ÕF,K>j~^--K'^~õh;+~õ^h^iE; ( 3 2 1 )

V. Nodal Flux Computation

After convergence of the iterative solution has been achieved, the fluxes, q,along the head-type boundaries are determined by substituting the solution backinto the global matrix equation for the rows that correspond to specified nonzeroflux nodes.

W"1-^) <3-22>

13

FINITE ELEMENT MODEL VERIFICATION

A computer code called "INFIL-FLOW" has been developed to simulate unsteadytwo-dimensional water flow in variably saturated porous media. The code isbased on the Galerkin finite element formulation and the solution techniquesthat were described in previous chapters. To verify the code, three illustrativeexamples are presented. The examples are chosen from literature for thepurpose of comparison.

I. 1 -D Transient Vertical Flow

Geometry and boundary conditions of the problem are depicted in Figure 1. Aflux of q - 3 cm/day is specified at the top boundary. The soil column has a depthof 150 cm with the water table located at the bottom where the boundarycondition is the prescribed pressure head, h=0. Fluxes, q = 0, are specified alongthe two vertical sides of the domain. The soil properties for loamy soil aredescribed by van Genuchten's model (1980a) with this data: 8, = 0.10,6S - 0.45,À = 0.025 cm"1, v - 1.65, and Kg = 50 cm/day. The initial pressure headdistribution is taken to be the steady-state solution of 1 -D Richards equationwijh zero flux at the top boundary and zero pressure head at the water table.

The results are compared in Figure 1 with that obtained by van Genuchten's 1 -Dfinite element model (1980b). The deviations in moisture are about 1.5 percentnear the top of the soil column. Agreements improve for the lower depths.

II. 2-D Transient Flow

Selim and Kirkham (1974) solved the 2-D diffusion-type flow equation forunsaturated soils using a finite difference method. With some modification tothe "INFIL-FLOW," the same governing equation was solved by "INFIL-FLOW"using the same geometry and soil conditions. The governing equation in terms ofsoil moisture 9 is

and soil water diff usivity, D(0), and hydraulic conductivity, K(6), are expressed as:

0(0) = 3.33 x K r V 9 ' 3 4 0 ^ 2 / min

K(0)« 3.33 x 1CT5e54-11*c/r7/ min ( 4 2 )

Initial water content is 0.20. The geometry and boundary conditions are shownin Figure 2 in which the results from both the Selim and Kirkham's model andthe "INFIL-FLOW" are compared by lines of equal water content for time t = 10

15

min. and time = 50 min. For high water content contour lines, the comparison isexcellent, but the fit is not very satisfactory near the wetting front where the lowwater content contour lines are located.

III. Steady-state Flow through a Square Embankment

Geometry and boundary conditions of the problem are depicted in Figure 3.No-flow boundary conditions exist at the base and top of the embankment. Thewater levels on the left- and right-hand side boundaries are 10 m and 2 m,respectively. On the seepage face, the pressure head is equal to zero. Along theboundary above the exit point, the flux normal to the boundary is zero. The heightof exit point is an unknown and determined by trial and error procedures.Hydraulic properties are

(\h-ha\f

= 1 h ^ ha

h<ha (4.3)

(4.4)

(4.5)

where A - 10 m, B = 1, and n = 1 are parameters, ha - 0 is the air entry value(Huyakorn et al. 1986). Values of physical parameters used in the simulation aresaturated conductivity ks - 0.01 m/day, specific storage Ss - 10"4m"1 .saturatedand residual water contents 6S = 0.25 and 6, = 0.2. The results are compared tothose obtained by Huyakorn et al. (1986) shown in Figure 3. The solutions for thelocation of water table and the distribution of base pressure head at the base arealmost identical to those predicted by Huyakorn's model.

16

FOK'S TWO-DIMENSIONAL INFILTRATION MODEL

Fok et al. (1966, 1967, 1982) related infiltration rate to time. They found fromexperimental observations, that the relationship between length of wetting andtime can be expressed as an exponential equation. They observed that wheninfiltration rates are plotted against corresponding infiltration times on a log-logpaper, an exponential form is usually obtained. In this infiltration model, the soilis assumed to be homogeneous and unsaturated with a uniform initial moisturedistribution. Also, the structure of the soil is assumed not to change afterwetting. For two-dimensional flow, the loci of the wetting pattern are assumedto be half ellipses and the vertical and horizontal flow components are describedby one-dimensional infiltration.

Fok et al. developed an equation for one-dimensional downward water move-ment in soil during infiltration. Darcy's Law

L (5.1)

was equated with the equation of continuity.

^ (5.2)

The resulting equation was integrated to arrive at an expression for downwardsoil water movement, and was rearranged such that

-Y— - ¡n (1 + -X- ) - ^ (5 3)

in which

Q - flow rateA = gross cross-sectional area through which flow occursy - length of wetting from soil surface to wetting front during infiltration

hi - constant total head loss in the transmission zone extrapolated to thewetting front = ho + hc - hw

where:ho - constant depth of water on soil surfacehc - constant capillary potential head at wetting fronthw - constant pressure potential loss in wetting front in excess of

transmission losst - time of infiltration

The relationship between the two dimensionless parameters y/hT andtK/nshT of Equation (5.3) shown in a log-log plot is curvilinear as shown inFigure 4, The curvilinear relationships are represented by several straight lines.

17

Four consecutive straight lines were used to replace the curvilinear relationshipbetween y/hT = 0.01 and y/hT - 30.

Four power equations showing y/hT as a function of tK/nshT were developedfrom the straight line approximation.

For0.1<y/h=<0.1

hT = nshT

J (5.4)

For 0.1 <y /h T <

For 1 < y/hT < 5

And for 5 < y/hT

1

< 3 0

y .hT

v nshT

nshT

(5.5)

tK O.85) (5.7)

Instead of calculating the y/h i for each infiltration problem for which theseequations can be used, four times can be calculated to specify points at whichthe straight line approximations change. These typical times ti, t2, b, and t» for theseparation of different infiltration flow periods can be obtained by substitutingthe values of y/hî into the corresponding equations such that

Fory/hT = 0.1

For y/hT = 1

f 2 - 0.316 — ( 5 9 )

For y / h ï - 5nshT

'3 = 3 . 2 6 - ^

18

Fory/hT= 30

t = 26 86—-^-

The length of the wetting from soil surface to the wetting front during infiltrationcan be calculated directly by rearranging Equations (54), (5.5), (5.6), and (5.7) bymultiplying both sides by hi- From this manipulation, four equations for thedownward movement in soil are found.

For t < ti

KhTt 0 5

n s (5.12)

For t, < t < t2

y=1.82(- Tns * f55 (5.13)

Fort2<t<t3

Kh0A7ty-2.19( ns )• ( 5 1 4 )

For t3 < t < t4

0 8 5) (5.15)

Infiltration in the horizontal direction is simpler because the component ofgravity is zero and the water is drawn into the soil by matrix suction forces only.Many studies such as Green and Apt (1911) and Toscoz et al. (1965) haveexpressed the horizontal advance of the wetting front, x, as a function oft12. Fokand his colleagues followed the derivation of Hansen (1955).

Darcy's Law states

QX = KA^- (5.16)

and the continuity equation is

%-at

19

where:

Qx - horizontal flow ratehx - horizontal pressure head lossx = lateral advance of wetting front

ns - 68 = incremental volumetric moisture content

Equating the two equations yields

ÙL (5.18)

and integrating the equation leads to

x ,1/2,1/2 <519>X~K ns ) l

The model was placed into a computer program by Kim (1986). This programwas written to include effects of a capillary fringe in the soil caused by a watertable. Kim's program was revised for this study to include soils with or without acapillary fringe. The model, which is based on the concepts and theoriesdeveloped by Fok and his colleagues, was programmed by Kim and has beenmodified by the authors; it will be referred to as "Fok's Model" throughout thisreport.

20

LABORATORY MODEL DESIGN AND CONSTRUCTION

The laboratory model consisted of three major parts: the soil compartment,tailwater compartment, and infiltration trench (Figure 5). The soil compartmentheld the soil medium that was to be tested. The tailwater compartment was usedto establish a water table in the soil medium. The infiltration trench was simplyan 8-inch wide and 28-inch long rectangular box to which water was added.

First, the size of the model had to be decided upon. Fok, Chung, and Clark (1982)used a box with a 27.6-inch width for a continuous line source of water supply.Yim and Sternberg's experiment (1984) used a 78-inch width for a trapezoidaltrench with a 48-inch top width. These models did not experience any boundaryeffect from the side walls. The present laboratory model was constructed 56inches wide and 48 inches high. The trench used in this model was 8 incheswide. The dimensions used in this model were considered acceptable.

The model was constructed of 3/4-inch thick A/C-grade plywood treated with awater sealer that prevented water from being absorbed into the wood. All jointsin the tailwater compartment were sealed with a silicone caulk to preventleakage. The edges of the soil and tailwater compartments were reinforced withcorner brackets to prevent buckling. The tailwater compartment or outer boxwas constructed first and nailed to the bottom of the model. The inner box or soilcompartment was then placed inside the outer box, centered and bracketed tothe bottom of the model to keep it in place (Figure 6). The inner box was notcaulked because water was to flow freely from the soil compartment to thetailwater compartment. The entire model was placed on cinder blocks to allowfor easier drainage from the bottom valve.

The sides of the soil compartment had 1 /4-inch holes drilled in a 1 -inch grid toallow the water to flow freely from the soil compartment into the tailwatercompartment. The sides of the soil compartment were lined with filter fabric toprevent the soil from moving into the ta'ilwater compartment.

The tailwater compartment was used to establish a water table in the soilmedium. The tailwater level or water table height could be controlled by sixvalves located at different elevations with 6-inch intervals around the sides ofthe box (Figure 7). All of the outlets were 1/2-inch valves.

The infiltration trench was constructed by building a frame and then staplingaluminum sheeting to the frame. The aluminum sheeting was punched with1/16th-inch diameter holes in a 1/3-inch grid. This allowed the water to flowfreely from the trench but at a uniform, controlled rate.

21

The water content, 8, at different locations in the soil medium was measured bysoil moisture meters purchased from the SoilMoisture Equipment Corporationof Santa Barbara, California. These water contents were measured to track theflow of the water and the position of the wetting front in the soil medium. The soilmoisture meter worked by measuring the resistance in ohms in the soil moistureblock. The resistance corresponded to specific meter readings. Both theresistance and meter readings can be related to the soil suction in bars (Figure 8).For each soil in which the soil moisture meter and blocks were used, a graphcould be produced relating the soil moisture meter readings to the watercontents (Figure 9).

Therefore, the soil moisture meter readings had to be calibrated with the watercontents of the soil being tested. This was done by placing a known volume of drysoil in a glass container. A small volume of water was then added to thecontainer. From these volumes, the volumetric water content of the soil could becalculated. When the water was evenly distributed throughout the soil in theglass container, the soil moisture meter was read giving the correspondingreading to that specif ic water content. Evaporation of the water f rom the soil wasprevented by sealing the container. This experiment was done twice for each soilto verify the results. From this data a graph was plotted for each soil relating thevolumetric water content to the soil moisture meter readings.

Porosity of the soil was also found from the above experiment. Porosity is definedas the volume of voids to the total volume and is expressed as a decimal orpercentage. When the soil became saturated, the volume of water was equal tothe volume of voids and the porosity could be calculated.

Three different soils were tested in the laboratory model. The first was a concretesand; the second, a clay loam; and the third, a loam. The permeability of the sandwas measured with a falling head laboratory apparatus. The permeabilities ofthe clay loam and the loam were measured in the soil box with a GuelphPermeameter purchased from the SoilMoisture Equipment Corporation. Thetests in both cases were run twice to verify the results. The Guelph Permeameteris an in-hole constant-head permeameter using the Mariotte Principle. Themethod involves measuring the steady-state ra.te of water recharge into the soilfrom a cylindrical well hole in which a constant head of water is maintained(Figure 10). Constant head in the well hole is established by regulating the levelof the bottom of the air tube. As the water level in the reservoir falls, a vacuum iscreated above the water and is relieved when air bubbles emerge from the airinlet tip and rise to the top of the reservoir. When a constant well height of wateris established, a bulb of saturated soil is created (Figure 10). The shape of thebulb is numerically described by a C factor as shown in Figure 11. Once the bulbis established, the outflow of water from the well reaches a steady-state flowrate that can be measured. The outflow rate, diameter of the well and height of

22

water in the well are used to determine the hydraulic conductivity or pernieabHityby these equations:

K = G^QQ —* ^ iO i (6.1)

where

G2 - ?&- « 0.0041TT[2 H ^ J (H2 - HO + a2 (H^a - Ha^)] (6.2)

(6.3)

(6.4)

<?2 = (X)(/?2)or (/)(*,) (6.5)

in which

K = hydraulic conductivity in cm/secH1H2 = well height for first and second measurements, respectively, in cmC1C2 = C factors corresponding to Hi/a and H2/a, respectively

a = well radius in cmX = 35.8 cm2 = reservoir constant used when the inner reservoir only is

selected, expressed in cm2

Y = 2,14 cm2 = reservoir constant used when the inner reservoir only isselected, expressed in cm2

RIRÍ = steady-state rate of fall corresponding the Hi and H2, respectively, andconverted to cm/sec

For the clay soil, the inner reservoir only was selected and the equation becomes

K = (0.0041)(2.14)(/?a ) - <0.0054)(2.14)(/?1 ) (6.6)

The readings Ri and R2 were found at 5-cm well head height (Hi) and 10-cm wellhead height (H2), respectively.

A sieve analysis of each soil was performed according to the American Society ofTesting Materials (ASTM) standards. The sieve analysis aided in the classificationof the soils according to the triangular soil classification chart developed by theU.S. Department of Agriculture.

23

After the soil properties had been tested, the infiltration trench experiment wasready to be run. The soil was placed in the box and the soil moisture blocks werelayered in a grid. The trench was then placed in the center of the soilcompartment and the experiment was begun.

24

EXPERIMENTAL RESULTS ANDCOMPARISON WITH NUMERICAL MODELS

I. Sand Experiments

The first laboratory experiments were run with a coarse construction sand as amedium. The sand was placed in the soil compartment in layers up to a totaldepth of 37 inches. A sieve analysis was performed on the soil with the resultinggrain size distribution curve shown in Figure 12. From this graph, the medianparticle size, Dso, and the coefficient of uniformity, Cu, were found to be 0.6 mmand 3.5 mm, respectively. The sand was evenly compacted by leveling when thesoil moisture blocks were placed in layers. The sand was used for a few trialexperiments to gauge how much water should be used and at what rate. Soilmoisture blocks were arranged in six rows which were placed 6 inches apart andin seven columns that were 3-1 / 2 inches apart. The upper two rows containedonly five columns because the trench was imbedded in the sand to a depth of 8inches and the two middle columns were directly beneath the trench (Figure 13).A water table was established in the sand at a depth of 1 foot above the bottom ofthe soil compartment.

The soil moisture blocks were calibrated with the sand as described previously.The water content versus meter readings graph is shown in Figure 14. From thissame experiment, the porosity was found to be 35 percent. The permeability ofthe sand was determined in the soils laboratory using a falling head perme-ameter and was determined to be 0.945 inches per minute.

A capillary fringe formed in the sand just above the water table. The measuredinitial condition of the sand before water was added to the infiltration trench iscompared with the initial condition predicted by the finite element model asshown in Figure 15. The authors found the agreement to be excellent.

The equal water content line of B - 0.2 is plotted and compared with the finiteelement model results (Figures 16 and 17). In the sand, the water did not travel agreat distance horizontally. The wetting front simply merges downward to thecapillary fringe after 10 minutes of the experiment and the water moved rapidlydownward.

A second experiment was run in the sand to check whether the results could beduplicated (Figures 18 and 19). Both of these experiments were run with 4inches of water in the infiltration trench which was allowed to infiltrate the sand.Results in the second experiment were similar to those in the first. The waterinfiltrated the sand quickly and the wetting front did not move a great distance inthe horizontal direction.

25

These experiments show that sand ¡s a good medium for an infiltration trenchfacility because the runoff would drain through it quickly. This occurs because ofthe sand's high permeability. However, the sand experiments did not produce aclearly defined wetting front, which made comparison with the computermodels more difficult.

The Fok's model computer program was run with the initial soil moisturecondition in the sand due to the established water table and the following soilparameters: permeability, K - 0.945 inches per minute and porosity, n = 0.35.The head loss, h, was found in Panikar (1977). An h - 4.25 inches was used.

It was difficult to compare Fok's model to the sand experiments because themodel could not simulate the initial condition. Also, the permeability of the sandwas so high that the time limit (u) in Equation (5.11) for the'sand was 28minutes. Fok's model also predicted movement in the vertical direction muchgreater than the lower boundary limits of the soil compartment. For this reason,the Fok's model results are not compared with the lab results in the figures. Itcan be concluded from these experiments that Fok's model is not accurate forsands.

The finite element results agreed with the experimental results. This modelseems to be a good predictor of infiltration.

II. Clay Experiments

The clay was obtained from a construction site on the Virginia Tech campus.Permeability was tested in the soil box with the use of a Guelph Permeameterand was determined to be 3 x 10 3 inches per minute. The clay was calibratedwith the soil moisture blocks, resulting in a graph of water content versus themeter readings (Figure 20). A sieve analysis was performed on the clay toclassify the soil. The grain-size distribution curve of the clay is shown in Figure21. Using the triangular soil classification chart, the clay was classified as a clayloam.

The clay was sifted through a screen with 1 /4-inch square hole size. This wasdone to prevent the larger clumps of soil from being placed into the soilcompartment because it was probable that clumps would interfere with evenpropagation of the wetting front.

The clay was placed in the soil compartment in layers up to a total depth of 28.5inches. The trench was placed on top of the clay and embedded to a 1 /2-inchdepth. No water table was established in the soil compartment.

The clay was evenly compacted by leveling when the soil moisture blocks wereplaced in rows. The soil moisture blocks were placed in four rows. The lower row

26

was placed one foot from the bottom of the model. The lower three rows wereplaced six inches apart and the top two rows were three inches apart as shownin Figure 22. A water table was not established in these experiments becausethere was not enough depth to establish the capillary fringe zone. Without thewater table, it would also be easier to see the wetting front develop. The clay wasdry when placed in the box with an overall water content of zero.

The first experiment in the clay produced a well-defined wetting front. Theexperiment took a day to run because the wetting front moved slowly throughthe clay. The experiment was run with one inch of water in the trench, whichwas kept constant throughout the experiment. The clay experiment was runwith less water in the trench than the sand experiment. This was done becausethe permeability of the clay was so much less than the permeability of the sand.The wetting front developed more slowly so that the computer models could bebetter tested against the laboratory results.

The results of the first experiment are shown in Figures 23 and 24. The results ofthe first experiment compared well with the predictions of Fok's model. For Fok'smodel, a porosity of 48 percent and a permeability of 0.003 inches per minutewere used based on the laboratory test results. An h of 10.3 inches was usedand was found from Panikar (1977). Fok's model was a good predictor of thepropagation of the wetting front in both the horizontal and vertical directions.The first experiment results also compared well with the finite element modelresults.

The second clay experiment was run in the same manner. The results of thesecond experiment are shown in Figures 25 and 26. The results were a littledifferent in that the wetting front moved more in the horizontal direction. Apossible reason could be that the clay, after the first experiment, had to be driedand crushed and then placed back into the box. The finest particles of the soilwere placed on top of the soil medium and, therefore, the water could movemore easily in the horizontal direction instead of the vertical direction. Theresults of the second experiment were also well in line with Fok's model results.

Both experiments compared well with the finite element model predictions. Thecomputed and measured fluxes were very close. The measured flux was 0.5inVmin and the computed flux was 0.45 inVmin. Both Fok's model and thefinite element formulation appear to be good predictors on infiltration in clay.

III. Loam Experiments

A topsoil was obtained from a construction site in the town of Blacksburg,Virginia. The permeability of the soil was tested in the soil box with the use of aGuelph Permeameter and was determined to be 0.007 inches per minute. Thetop soil was calibrated with the soil moisture blocks resulting in a graph of water

27

content versus the meter readings (Rigure 27). A sieve analysis was performedon the topsoil to classify the soil. The grain-size distribution curve of the topsoil isshown in Figure 28. The topsoil was classified as a loam, using the triangular soilclassification chart.

The loam was sifted through the same screen used for the clay. The loam wasplaced in the soil compartment in layers up to a total depth of one foot eleveninches. The trench was placed on top of the clay and embeded to one-inch depth.The loam was evenly compacted by leveling when the soil moisture blocks wereplaced in four rows. The bottom row was placed six inches apart and the top tworows were three inches apart as shown in Figure 29. A water table was notestablished in the loam experiment because the depth of the loam was tooshallow. The loam had an average water content of 8 percent when theexperiment was begun. Only one experiment was performed on the loam.

The loam experiment, like the clay experiments, produced a well-definedwetting front. The experiment was run for 130 minutes. The wetting front in theloam moved more quickly than that of the clay and more slowly than that of thesand. The experiment was run with one inch of water in the trench and thewater level was held constant throughout the experiment. The results areshown in Figures 30 and 31.

The loam experiment compared relatively well with Fok's model. An h of 13.06inches was used and found from Panikar (1977). The vertical propagation of thewetting front was predicted well by Fok's model for 60 minutes. It did not predictthe horizontal propagation as well. After 70 minutes, Fok's model resultsexceeded the limits of the laboratory model and are not shown on the figures.

The loam experiment also compared well with the finite element modelpredictions. Like Fok's model, the finite element model predicted wetting frontmovement beyond the soil box boundaries at around 70 minutes. For thisreason, both results of Fok's model and the finite element model are comparedwith the laboratory model results up to 60 minutes.

28

PARAMETRIC STUDIES

Based on the comparison with the three numerical models and the laboratorytests, the finite element model developed in this study has been well verified. Itsutility will be demonstrated by simulations for variably saturated flow problems.It provides the necessary tool to study the behavior of water movement from aninfiltration trench to its surrounding porous medium.

To find the storage of water in the trench, routing is performed based on thefollowing continuity equation.

where

dS/dt = rate of change of storage S in the trenchI = inflow

O = outflow across the trench boundary into the surrounding porousmedia.

Equation (8.1) can be written in the finite difference form.

So — Si //> + At Oo — O*

Aí

where subscripts 1 and 2 refer to the previous and current time levels.

(8.2)

Before the soil at the trench bottom is saturated, the boundary condition is thespecified flux, which is equal to the inflow to the trench. In this case, all inflowinfiltrates the soils. The routing is actually not required because there is nostorage in the trench. The routing procedure is outlined:

Step 1. Assume a value for h(xB,yB,t)Step 2. Compute O2 through finite element analysisStep 3. Compute S2 by Equation (8.2)Step 4. Compute water level based on trench geometryStep 5. Check water level (computed h(xa,yB.t) against assumed value for

h(xb,yb,t)Step 6. If the difference beween the computed and the assumed h(xB,VB,t) is

negligible, go to the next time step. If the difference is not negligible,go to Step 1.

29

I. Soil Properties

To study the effects of soil properties on infiltration, the finite element model hasbeen applied to an infiltration trench, subjected to a given inflow hydrograph, butfor several different soils. The geometry of the trench and the boundaryconditions prescribed are shown in Figure 32. Soil properties are defined by vanGenuchten's model. The values used in the model are listed in Table 1 (Carseland Parrish 1988).

A wide variety of soils has been examined, ranging from silty clay to sand. Thetime steps used are 0.25 min. for sandy soil, 0.50 min. for loamy soil, and 1.00min. for silt and clay. Water table isf ixed at a depth 8 feet below the bottom of thetrench. Initial water content varies for each type of soil. The simulation resultsare shown in Figures 33 and 35. The inflow hydrograph to the trench is shown inFigure 33. The outflow due to infiltration from the trench to surrounding soils isshown in Figure 34 for different soils. The change of water level in the trench fordifferent soils is shown in Figure 35.

One observation from those results is that the saturated conductivity KU andparameter A in van Genuchten's model are the most important factors in thisinfiltration study. It has been shown from their model that large Ks and small Awill yield a high infiltration rate. Large HU is usually associated with large A.Infiltration rate is not simply proportional to the saturated hydraulic conductivity,but rather to the ratio of KU/M. Soils having large values of Ks/^ give highinfiltration rate and therefore the water levels in the trench are lower.

II. Geometries

The effects of geometries of the trenches on infiltration have been studied. Adeep, narrow trench is compared to a wide, shallow one for the same inflowhydrograph, trench length, and soil type. Sandy loam and loam are examined inthis study. Keeping the total trench volume constant, deep trenches give ahigher infiltration rate because of the increase of pressure head due to highwater depth in the trench. However, wide trenches give higher infiltrationvolume due to the large horizontal area through which vertical infiltration takesplace. Both the width and depth of the trench play important roles in thedetermination of infiltration volume.

Figures 36 and 37 show that wide, shallow trenches give a higher infiltrationrate initially and approach the same steady-state infiltration rate. Comparisonhas also been made for the two trenches which have the same input hydrographand trench depth but are different in length and width. The trench volumeremains constant. Results shown in Figures 38 and 39 indicate that the short,wide trench has a higher infiltration rate and lower water level in the trench thanthe long, narrow trench.

30

III. Water Tables

Because of the capillary effect, the location of the water table will determine theinitial water content distribution. For different soils, the effects of the water tablelocation vary. The model is applied to a one-foot long, one-foot deep, andtwo-foot wide trench filled with water all the time. The infiltration rates at theend of the first minute have been computed for different water table locations.Results for different soils with a change of water tables from three foot to 80 footare shown in Figure 40. There exists a limit for each soil beyond which the watertable has no effect on infiltration. The limit is defined as the depth at whichchange in infiltration rate is less than one percent of the infiltration rate at theprevious depth. The limit for sand is about 10 feet; 30 feet for loamy sand; and 60feet for sandy loam.

IV. Effects of Sediment Deposition in Trench

Sediment deposition will reduce the permeability on the trench bottom. A 3-inchthick sediment layer is assumed with 75 percent, 50 percent, and 25 percent ofthe original permeability, respectively. Using the same trench geometry andinflow hydrograph as shown in Figures 32 and 33, routings are performed for asandy loam soil. The results are shown in Figure 41. There are obviousreductions in infiltration rate before the maximum water level in the trench isreached. After this point, the differences become small in infiltration rates forreductions in permeability which are less than 50 percent.

V. Distribution of Water Content

A 2,000-minute infiltration process has been simulated to examine the move-ment of the wetting front due to the rising and falling of water level in the trenchduring a storm event. The trench geometry shown in Figure 32 and the inflowhydrograph shown in Figure 33 are used. The soil chosen for this simulation isloam. In figures 42 and 43, the advancing of the wetting front due to soilmoisture movement and the change of water levels in the trench are shownusing the results predicted by the finite element model.

31

CONCLUSIONS

In this study, a two-dimensional finite element model has been developed tosimulate the transient flow of water in a variably saturated porous medium. Themodel simulates pressure heads and moisture distributions in the soilssurrounding an infiltration facility and fluxes across the boundaries of thefacility. Routing has also been performed to find the infiltration rate, water depth,and storage in the facility, and any overflow that may occur.

Governing equations have been derived and placed into a finite element analysisto arrive at the finite element formulation. The finite element model has beenverified for one-dimensional transient vertical flow by comparison with vanGenuchten's finite element model. It has also been verified for two-dimensionaltransient flow by comparison with the Selim and Kirkham model. Lastly, themodel has been verified for steady-state flow through a square embankment bycomparison with the results predicted by Huyakorn et al.

The results of the finite element formulation andan infiltration model developedby Fok et al. have been compared with the results of laboratory model study.Fok's model is a two-dimensional infiltration model for furrow irrigationexpressed in explicit form. Vertical and horizontal advances of the wetting frontare calculated based upon Darcy's Law and the principle of continuity. Thelaboratory model was designed to test infiltration trenches specifically and usedsoil moisture blocks to measure the water contents in a grid pattern within thesoil being tested. Three soils were tested. The results of soil moisture distributioncompared well with the predictions of the finite element model for all three soils.In this study, Fok's model was not accurate for the sand, but was reasonablyaccurate for the clay and the loam. The finite element formulation developed inthis study has been well verified by comparisons with other numerical models,Fok's infiltration model, and laboratory tests.

Parametric studies have been performed with the finite element model for loamsoil and sandy loam to determine the various effects of such parameters as soilproperties, facility geometries, and depth of water tables. Sands are recom-mended for use in an infiltration facility while clays are not. Saturated soilconductivity and the /\ parameter from van Genuchten's model are found tohave the greatest effect on infiltration rate. Where geometry is concerned, awide, shallow trench is recommended for a given trench length and trenchvolume. A short, wide trench is recommended for a given depth and trenchvolume. The depth of the water table has been shown to have a greater effect insilt than in sand. There exists a limit for each soil beyond which the water tablehas no effect on infiltration. The limit for sand is about 10 ft.; 30 ft. for loamysand; and 60 ft. for sandy loam.

33

In the design of an infiltration trench, geometries vary considerably and waterdepth in the trench varies with time. The water flux across the bottom and sidesof the trench is an important factor in sizing the trench. The finite elementformulation has the capability to change geometries and is a useful tool in thedesign of infiltration facilities to calculate the flux and water level in the trench.

34

FIGURES

FIGURE 1Comparison of Finite Element Model with van Genuchten's Model

q = 3 cm/day

water table

150.00

125.00 -

100.00 -

Õ 75.00 -

50.00 -

25.00 -

0.000.00

initial conditionINRL-FLOW (t=0.5 day)V-G Modle (t=0.5 day)IIMFIl-FLOW (t=1.0 day)V-G Model (t=1.0 day)

0.10 0.20 0.30moisture

0.40 0.50

37

FIGURE 2Comparison of Finite Element Model with Selim and Kirkham's Model

10 minutes 50 minutes

U 40 cm

' soil surfoce

. Selim and Kirkham's ModelINFIL-FLOW

38

FIGURE 3Comparison of Finite Element Model with Huyakorn's Model

y

/=1

0

m

•

lU In

q=o

• *

h s

exit point

Vh+y=2 m

10.00

8.00 -

6.00 -

4.00 :

2.00 -

water table

base pressure head

INFIl-FLOWHuyakorn's Model

0 . 0 0 | I i I I i i i i i | i T r T i i r i i ; i i i i i i r I r '| I l M I i i i i | i

0.00 2.00 4.00 6.00 8.00 10.00x (m)

39

FIGURE 4Relationship between y/h and Kt/nsh

(Fok et al. 1982)

0.01

40

FIGURE 5Top View of Laboratory Model

Tailwater compartment

Soil compartment

8"

L

28"

Infiltration trench 48" b6"

48"

56" -31

41

FIGURE 6Three-dimensional View of Laboratory Model

Panel C

Panel D

threaded holes

56'

0

42

FIGURE 7Panels of Tailwater Compartment with Outlets

Panel A

T2'

O

P

1/2" threaded holes

56"

Panel B

T2'6"

IT4> 6"

O

O

1 / 2 " threaded holes

Panel C

T.11

Panel D

O 1 / 2 " threaded hole

T O 1 /2 " threaded hole

4'

43

FIGURE 8Relationship of Meter Readings to Soil Suction

(SoilMoisture Equipment Corp. 1985)

o Q K -

40K-

30K-

mm +.

z| 800-

$ *°°400-

4 0 -

20-

.1

1

f/

/

t 4

y— ~\m

/

/

yM\f\

ÂA

> .i

sON.

...

1 1

SUenow »

A

I—

1 IMS

—

Ay

> i

- 5

-3D

- M

0

44

FIGURE 9Relationship of Meter Readings to Water Content

(SoilMoisture Equipment Corp. 1985)

fïssïï

1

ï

1sa

5•5

100

« 0

8 0

70

eo

so

4 0

30

20

RtKTIonMilp of rMdlngt on Mwxl9201 SollaoUtur* Slockt to n i l•olitur» contant «ntf toll luetlon•iwn n M »lth ttxtol Ï»IO-A Soll-«olirui-» mfmr.

ABCX)

OtVMNC «AT

VOID t l» t tOAM

HAHrOU IAN»

IHMO «»N»Y iOAHI

45

FIGURE 10Guelph Permeameter

(SoilMoisture Equipment Corp. 1986)

Outer ReservoirTube

Water Supply inReservoir Tube

If height of water in welldrops below level of Air InletTip, air wil l bubble intoreservoir until water inwell rises and shuts offair supply.

The Mariotte Principleprovides that:

Partial Vacuum, P

Standing Columnf W P

Water in well is at samelevel as Air Inlet Tip

PermBameter

Water in Well

Zone of Saturation

46

FIGURE 11C Factor for Guelph Permeameter

(SoilMoisture Equipment Corp. 1986}

C3.0

•C" Factor

1 S anda2 Structured loams and clays3 Unstructured clay*

2X1

1.0

5H/a

10

CO

too

70

60

w so

u 30

10

1000

FIGURE 12Grain Size Distribution Curve of Sand

U. S. STANDARD SIEVE SIZE

1.5" VA" 3/Í" Me.* 10 10 « 60 100 2O0

1 T1

Í

warn

1 \

¡1 \M '

1 N1 \i1i

i iV 'N'\'\\\\1 ^

i

\

ji

iii

iiii

\ i

•

—

100 10 0.1CHAIN SHE I» MILLIMETERS

GRAVEL SAND

I rim CO* Kit 1 WCPrtlM I MHE

0.0) 0.001

SILT OR CLAT

FIGURE 13Placement of Soil Moisture Blocks in Sand

6"

o o o o

o o o o

4-

¿

6"

-* o6"

oto o o o o c b

o o o o o o

H- O O O O O O O

49

FIGURE 14Water Content versus Meter Readings for Sand

Meter Readings

50

FIGURE 15Initial Moisture Condition for First Sand Experiment

3'1*i

Y.

20"

i i n

= 20%

FEM — D — • —

measured + + *-*-+ *-

51

OÍf

FIGURE 16Results of First Sand Experiment (10 and 30 Minutes)

FEM — | |— measured - f + 4- •+•

FIGURE 17Results of First Sand Experiment (50 and 100 Minutes}

3'r

CO

FEM — I | — measured + +

en

FIGURE 18Results of Second Sand Experiment (Initial Condition and 20 Minutes)

FEM — D ~ measured —|— —|— —|— —|—

FIGURE 19Results of Second Said Experiment (50 and 100 Minutes}

en

0 = 20% ++ •

3'1*

FEM — Q — measured + +

FIGURE 20Water Content versus Meter Readings for Clay

au

i.8

Meter Readings

56

FIGURE 21Grain Size Distribution Curve of Clay

l i . S. STANDARD SIEVE SIZE

3" 1.5" 1/*" 3/8™ No. 4 10 20 40 60 100 200

I 70

au 30QL.

20

10

- — -

—

1

—

1

f

f ••-"f -

-ii

. . ....i

_J _ ...

N

_ i

i

—

i

S

V

—

-

.— .

-

-

....

—

— • • -

!000 100 10 1.0 o.t 0.01

COBBLES GRAVELCOARSE | FINE

GRAIN SIZE

COARSE

IN

1

MILLIMETERS

SANDMEDIUM 1 FINEH

en

FIGURE 22Placement of Soil Moisture Blocks in Clay

o o o o oo o o o o o fo o o o o o ó

o o o o o o ó43'

1'

58

2'4V2"

FIGURE 23Results of First Clay Experiment (37 and 57 Minutes)

37 minutes

e 20"

e = 20%

2'414"

9=20%

Fok- measured + + +

CO

O)

o FIGURE 24Results of First Clay Experiment (147 and 237 Minutes)

2V/2

147 minutes

20"

9 =20%

•A—•

Fok A FEM Q

measured + +++-H-

237 minutes

20"

4 / 2

FIGURE 25Results of Second Clay Experiment (39 and 84 Minutes)

39 minutes

20"

e =20%

24 y2"

O)

Fok A FEM •

measured + + + + +

0=20^-/o

toFIGURE 26

Results of Second Clay Experiment (164 and 224 Minutes)

2'4'A

164 minutes

20 -H v

0 = 20%

J*-4"-»|

VÜ,

2'4'

Fok — A — FEM Q measured •+ + + +

FIGURE 27Water Content versus Meter Readings for Loam

T50

20

COo-O 10 20 30 40 SO 60 70 00 90 100

Meter Readings

O)FIGURE 28

Grain Size Distribution Curve of Loam

U. S. STANDARD SIEVE SIZE

1.5" 3/*" 3/S" No. 4 10 Î0 40 «0 100 200

90

80

70

60

Ul JO

. . . .

—- —

1— • 1 1

1 .

-•[••-

1

>

7 T~

1

— I

\ FNI¡i|

—

s\

-

—

— -

—

—

- •

—

1000 100 10 1.0 0.1 0.01

COBSLES CNAVELCOURSE ¡ FINE

GRAIN SIZE IK

COASSE 1

MILLIMETERS

SANOMEDIUM [ FINE

SILT OR ClAY

FIGURE 29Placement of Soil Moisture Blocks in Loam

1"

t

Jr2"

3»*

6"

L̂

16"

O

o

o

o

oo

o

o

oo

o

o

oo

o

o

o

o

o

o

o

o

65

OÍ

o>FIGURE 30

Results of Loam Experiment (2 and 10 Minutes)

I ' l l *

e = 20%

I ' l l "

10 minutes

2 0 "

K—

hjAe = 20%

Fok — A — FEM - measured + + + +

FIGURE 31Results of Loam Experiment (25 and 60 Minutes)

Fok — ^ -FEM— Q meausred -f- + •+•

en

FIGURE 32Geometry of the Trench for Parametric Studies

q = 0

8 ft

16 ft

68

FIGURE 33Inflow Hydrograph to the Trench Prescribed for Parametric Studies

0.60

c

'0.40 -

Z5O

o

0.20 -

0.000.00 40.00 80.00

time (min)120.00

69

FIGURE 34Infiltration Rates for Different Soils

0.40

.0.30 H

3 0.20

co

0.10 H

0.00

o-e-6-B-B s o n c |

loamy sandsandy loamloamsilty loamsilt

o.oo 40.00 80.00time(min)

120.00

70

FIGURE 35Water Levels in the Trench for Different Soils

8.00

7.00 :

6.00 ~

,5.00 :

0.00

loamy sandsandy loamloamsilty loomsilt

0.00 40.00 80.00 120.00time (min)

71

FIGURE 36Effects of Deep/narrow vs. Shallow/wide Trench on Infiltration Rates and

Water Levels in Trenches with Constant Volume (Sandy Loam Soil)

0.20Í

60 80 100 120

lime (mtn)— InflltratlonfdMp/narrow trench)— Inflltratlonishallow/wlde trench)— wafer l*velfda*p/narrow trench)—-- water l*v«lf shallow/wide trench)

72

FIGURE 37Effects of Deep/narrow vs. Shallow/wide Trench on Infiltration Rates and

Water Levels in Trenches with Constant Volume (Loam Soil)

0.20

100f îme (min)

• Infllrraffon(de*p/narrow trench)— Ihflltrattonfshallow/wlde trench)— water l«v»li deep/narrow trench)—•water leveKshallow/wtde trench)

120

73

FIGURE 38Effects of Long/narrow vs. Short/wide Trench on Infiltration Rates and

Water Levels ín Trenches with Constant Volume (Sandy Loam Soil)

0.20

0.00*20 60 80 100 120

lime (mtn)' ínflltraHorrflong/nqrrow frenen)• infNtratlorushorr/wldft trench)

LEGEND —- water l«v«l|'long/narrow trench)• water l*v«l short/wide trencrA

74

FIGURE 39Effects of Long/narrow vs. Short/wide Trench on Infiltration Rates and

Water Levels in Trenches with Constant Volume (Loam Soil)

0.20Í

0.0040 60 SO

lime (mtn)— Irrflltratlonftong /narrow trench)— Inftltrattonfshort/wïd* trench)—— water levelTlong/narrow trench)— water levl*(shori/wlde trenchi

120

75

FIGURE 40Effects of Water Table on Infiltration Rate

0.15

0.10 -

13

§0.05

0.00

sandloamy sandsandy loamloamsilt

0.00 20.00 40.00depth (ft)

60.00 80.00

76

FIGURE 41Effects of Sediment Deposition in Trench on Infiltration Rate

0.15

0.10 -

O)

o

g 0.05

° coa-e 100s? of original K° ° ° ° ° 7553 of original Kooooo 5Q% of original K

25 of original K

0.000.00 40.00

time (min)80.00 120.00

77

FIGURE 42Long-term Simulation of Infiltration in a Trench

0.03

0.02

0.01

0.00

13 i infiltration

kwater level

/"* flux across

Trrr-water table

8

7

$

5?

300 600 900 1200 1500 1800 2100

"time (mfn)

78

FIGURE 43Movement of Wetting Fronts in Soil Medium as a Function of Time

Q Q O ° ° 60 min120 min240 min480 min960 min2000 min

79

TABLES

TABLE 1Typical Values of van Genuchten's Model

Soil

ClayClay LoamLoamLoamy SandSiltSilty LoamSilty ClaySilty Clay LoamSandSandy ClaySandy Clay LoamSandy Loam

Kslcm/h)

0.200.261.04

14.600.250.450.020.07

29.700.121.314.42

0s

0.380.410.430.410.430.450.360.430.430.380.390.41

e,

0.070.090.080.060.030.070.070.090.040.100.100.06

0.0080.0190.0360.1240.0160.0200.0050.0100.1450.0270.0590.075

V

1.091.311.562.281.371.411.091.232.681.231.481.89

83

REFERENCES

American Society of Agricultural Engineers. 1983. Advances in Infiltration. Conferenceproceedings, Chicago.

Beukeboom, T.J. 1982. "A Dry-Well System for Excess Rainwater Discharge." Pro-ceedings of the International Symposium on Urban Hydrology, Hydraulics andSediment Control. Lexington, Kentucky.

Bouwer, H. 1978. Groundwater Hydrology. McGraw-Hill, Inc., New York,

Carsel, R.F., R.S. Parrish. 1988. "Developing Joint Probability Distribution of Soil of WaterRetention Characteristics." Water Resources Research. 24:5, 755-769.

Eagleson, P.S. 1970. Dynamic Hydrology. McGraw-Hill, Inc., New York

Ericsson, L.O., and G. Gustafson. 1982. "Design of Stormwater Recharge Basins."Proceedings of the Conference on Stormwater Detention Facilities. Henniker, NewHampshire.

Fok, Y., and V.E. Hansen. 1966. "One-Dimensional Infiltration into Homogeneous Soil."Journal of Irrigation and Drainage. ASCE, 92;3, 35-48.

Fok, Y. 1967. "Infiltration Equation in Exponential Forms." Journal of Irrigation andDrainage. ASCE, 93:4, 125-135.

Fok, Y., S. Chung, and C.K. Clark. 1982. "Two-Dimensional Exponential InfiltrationEquations." Journal of Irrigation and Drainage. ASCE, 108:4, 231-241.

Green, W.H., and G.A. Ampt. 1911. "Studies on Soil Physics.Part 1 —The Flow of Air andWater through Soils." Journal of Agricultural Science. Vol. 4,1 -24.

Hansen, V.E. 1955. "Infiltration and Soil Water Movement During Irrigation." SoilScience. 79:2,93-105.

Hantzche, iM.N., and J.B. Franzini. 1980 "Utilization of Infiltration Basins for UrbanStormwater Management." Proceedings of the International Symposium on UrbanRunoff. Lexington, Kentucky.

Higgs, G.E. 1978. "A Case Study of Percolation Storage." Proceedings of the InternationalSymposium on Urban Stormwater Management. Lexington, Kentucky.

Huyakorn, P.S., E.P. Springer, V. Guvanasen, and T.D. Wadsworth. 1986. "A Three-Dimensional Finite-Element Model for Simulating Water Flow in Variably SaturatedPorous Media," Water Resources Research, 22:13, 1790-1808.

Kaluarachchi, J.J., and J.C. Parker. 1987, "Finite Element Analysis of Water Flow inVariably Saturated Soil." Journal of Hydrology, 90:269-291.

Kim, J. 1986. A Study of Infiltration Trenches in Unsaturated Soil. Masters thesis,Virginia Polytechnic Institute and State University, Blacksburg.

85

Li, W. 1983. Fluid Mechanics in Water Resources Engineering. Allyn and Bacon, Inc.

Maryland Department of Natural Resources. 1984. Standards and Specifications forInfiltration Practices. Stormwater Management Division, Water Resources Admin-istration.

McBride, M.C., and Y.M. Sternberg. 1983. Storm Water Management InfiltrationStructures. Maryland Department of Transportation, State Highway AdministrationResearch Report, No. AWO83-253-046.

Northern Virginia Planning District Commission. 1978. Guidebook for Screening UrbanNonpoint Pollution Management Strategies.

Northern Virginia Planning District Commission. 1987. BMP Handbook for the OccoquanWatershed.

Panikar, J.T., and G. Nanjappa. 1977. "Suction Head at Wet Front in Unsaturated-flowProblems — A New Definition," journal of Hydrology. Vol. 33, 1-14.

Poertner, H.G. 1974. Practices in Detention of Urban Stormwater Runoff. AmericanPublic Works Association, Special Report No. 43.

Reddy, J.N. 1984. An Introduction to the Finite Element Method. McGraw-Hill, Inc.New York.

Schueler, T.R. 1987. Controlling Urban Runoff: A Practical Manual for Planning andDesigning Urban BMPs. Metropolitan Washington Council of Governments.

Selim, H.M., and D. Kirkham. 1974. "Unsteady State Two-Dimensional Water ContentDistribution and Wetting Fronts in Soils." Geoderma, 11, 259-274.

SoilMoisture Equipment Corporation. 1985. Operating Instructions for the Model 5201Soilmoisture Blocks.

SoilMoisture Equipment Corporation. 1986. 2800 Kl Operating Instructions.

South Florida Water Management District. 1983. Permitting Information Manual. Vol. IV,Management and Storage of Surface Waters.

Toscoz, S.D., D. Kirkham, and E.R. Baumann. 1965. 'Two-Dimensional Infiltration andWetting Fronts." Journal of Irrigation and Drainage. ASCE, 91:3,4477, 65-79.

Virginia Water Control Board. 1978. Best Management Practices Handbook.

van Genuchten, M.T. 1980a. "A Closed-form Equation for Predicting the HydraulicConductivity of Unsaturated Soils." SoilScience Society of'America Journal. 44,892-99.

van Genuchten, M.T. 1980b. "A Comparison of Numerical Solutions of the One-Dimensional Unsaturated-saturated Flow and Mass Transport Equations." Pro-ceedings of the Third International Conference on Finite Elements in Water Resources.University of Mississippi.

86

Yeh, G.T. 1987. FEMWATER.A Finite Element Model of Water Flow through Saturated-Unsaturated Porous Media. Environmental Science Division, Publ. 2943, Oak RidgeNational Laboratory.

Yim, C.S., and Y.M. Sternberg. 1984. Laboratory Tests of Stormwater ManagementInfiltration Structures. Maryland Department of Transportation, State Highway Admin-istration Research Report, No. AW084-288-046.

87